BGE python: accessing Edge information in a mesh made of polylines (not polygons)
The BGE does not know edges. The BGE knows polygons (faces) which are defined by 3 or 4 vertices.

An edge in this context is a semantic interpretation of a line between two vertices. It does not exist as object nor is it used anywhere. The render engine simply does not need it.

You need to know in the BGE the faces do not necessarily share the same vertices even when in Blender it seams like that.

I can give you a general guideline:


Iterate over the polygons of the mesh. 
A polygon has 3 or 4 vertices. An "virtual" edge are two consecutive vertices plus the edge between the last and the first vertex. You will get as many edges as the polygon has vertices
Now you should have all edges (with begin and end vertices) of all polygons.
You can match them to each other by comparing the vertex positions. Be aware edges have a direction. You need to consider the reverse of an edge when comparing.